A 2-variable oriented knot polynomial 
motivated by the Jones
polynomial (Freyd et al. 1985). Its name is an acronym for the last names
of its co-discoverers: Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter (Freyd
et al. 1985). Independent work related to the HOMFLY polynomial was also carried
out by Prztycki and Traczyk (1987). HOMFLY polynomial is defined by the skein
relationship
 |
(1)
|
(Doll and Hoste 1991), where 
is sometimes written instead of 
(Kanenobu and Sumi 1993) or, with a slightly different relationship,
as
 |
(2)
|
(Kauffman 1991). It is also defined as 
in terms of skein
relationship
 |
(3)
|
(Lickorish and Millett 1988). It can be regarded as a nonhomogeneous polynomial in two variables or a homogeneous polynomial in three variables. In three variables the skein relationship is written
 |
(4)
|
It is normalized so that 
. Also, for 
unlinked unknotted components,
 |
(5)
|
This polynomial usually detects chirality but does not detect the distinct enantiomers of the
knots 09-042, 10-048, 10-071, 10-091, 10-104, and 10-125
(Jones 1987). The HOMFLY polynomial of an oriented knot
is the same if the orientation is reversed. It is a generalization of the Jones
polynomial 
,
satisfying
 |  |  |
(6)
|
 |  |  |
(7)
|
It is also a generalization of the Alexander polynomial 
,
satisfying
 |
(8)
|
The HOMFLY polynomial of the mirror image 
of a knot 
is given by
 |
(9)
|
so 
usually but not always detects chirality.
A split union of two links (i.e., bringing two links together without intertwining them) has HOMFLY polynomial
 |
(10)
|
Also, the composition of two links
 |
(11)
|
so the polynomial of a composite knot factors into polynomials of its constituent knots (Adams 1994).
Mutants have the same HOMFLY polynomials. In fact, there are infinitely many distinct knots with the same HOMFLY polynomial (Kanenobu 1986). Examples include (05-001, 10-132), (08-008, 10-129) (08-016, 10-156), and (10-025, 10-056) (Jones 1987). Incidentally, these also have the same Jones polynomial.
M. B. Thistlethwaite has tabulated the HOMFLY polynomial for knots up to 13 crossings.
